ASTR 3730: Fall 2003

Neutron stars and pulsarsNeutron stars are formed in (some) core collapse supernovawhen an iron core becomes unstable to collapse. Basic properties:

• neutron rich because many of the protons have beentransformed into neutrons:

†

e- + p Æ n + n e

• degeneracy pressure of neutrons provides the pressuresupport against gravity, but matters are complicatedbecause nuclear forces are important.

• density is comparable to nuclear values: r ~ 1015 g cm-3,radius is ~10 km

ASTR 3730: Fall 2003

Newly formed neutron stars are very hot, and embedded within young supernova remnants. Rapidly cool due to neutrino emission (in the very early stages), then photonemission:

Galactic population ofneutron stars must beenormous - a very smallnumber have been detected directly from this cooling radiation.

ASTR 3730: Fall 2003

Neutron star historyIdea that neutron stars might be formed in supernova explosionswas suggested by Walter Baade & Fritz Zwicky in 1934. Detailed models for neutron stars were derived by Robert Oppenheimer and George Volkoff in 1939.

Obvious that isolated neutron stars would be very faintand hard to detect.Neutron stars in binaries are extremely luminous, but only in X-rays which at the time were undetectable.Surprisingly, first discovered by accident via pulsed radio emission - pulsars, by Jocelyn Bell in 1967.

ASTR 3730: Fall 2003

Large number (~ thousand) of radio pulsars are now known,some of which also pulse in optical or X-ray:

Crab pulsar

ASTR 3730: Fall 2003

Where neutron starmasses can be measured(in binary systems), themass is found to be veryclose to 1.4 Msun.

Maximum possible massfor a neutron star is notknown, but probably 2 - 3 Msun.

ASTR 3730: Fall 2003

Pulsar timingBasic observable for a radio pulsar is the time of arrival ofeach pulse at Earth. After correction for effects such as theEarth’s motion around the Sun, derive:

†

PdPdt

≡ ˙ P

d2Pdt 2 ≡ ˙ ̇ P

Pulse period (e.g. 1s, though a wide range)Period derivative (e.g. 10-12 seconds persecond - so dimensionless)

Second period derivative

Some (not all) pulsars make very good clocks - any trendsin P can be predicted / extrapolated to extraordinary accuracy.Frequency stability comparable to best atomic clocks.

ASTR 3730: Fall 2003

Isolated pulsars (ignoring some special cases we will discusslater) are powered by rotational kinetic energy. Spin down slowly as this energy is released:

†

Erot =12

IW2

dErot

dt=

12

I ddt

W2( ) =12

I ¥ 2WdWdt

= IW ˙ W

…where I is the moment of inertia of the neutron star and Wis the angular velocity. Very roughly:

†

I ~ MR2 take: M = 1.4 Msun, R = 10 km, W = 60 s-1

†

Erot = 5 ¥1048 erg (enough for a Solar luminosityfor ~40 million years)

Eventually, pulsars slow down so much that the pulsedradiation ceases.

ASTR 3730: Fall 2003

What can we learn from measuring P and its time derivatives?

Period and period derivativeQuantity:

†

P ˙ P has dimensions of time and is obviously related to the `age’ of the pulsar.

To be more quantitative, write:

†

dWdt

= -kWn

k is a positive constant, n is a constantcalled the braking index that describeshow the pulsar spins down. Simplest model for pulsar radiation predicts thatthe braking index n = 3.

Integrate this equation, assuming that:• at time t = 0, W = W0• at present time t = t, angular velocity is W

ASTR 3730: Fall 2003

†

W-ndW = -k dt0

t

ÚW0

W

Ú

W-n +1

-n +1È

Î Í

˘

˚ ˙

W0

W

= -kt

W-n +1

1- n-

W0-n +1

1- nÈ

Î Í

˘

˚ ˙ = -kt

Second term is negligible provided that:• W0 >> W i.e. pulsar was initially spinning

much faster than it is today• n > 1 so the powers are negative (actually

n = 3 or thereabouts so this is OK)

ASTR 3730: Fall 2003

Result is then:

†

-kt =W1-n

1- n

Eliminate k using the first equation for spin down:

†

˙ W = -kWn

†

t =1

1- nW˙ W

Since W = 2p / P can alternatively write this in terms of the period and period derivative with respect to time:

†

t =1

n -1P˙ P

e.g. for the Crab pulsar:

†

P = 33.4 ms, ˙ P = 4.21¥10-13 ss-1

Taking n = 3, this gives an age of t = 1255 years for the pulsar.Good estimate to the true age, since the Crab nebula was formed from a supernova in 1054 AD.

ASTR 3730: Fall 2003

Period plus first two period derivativesWith a measure of all three quantities, we can determine thebraking index without recourse to theory:

†

˙ W = -kWn Æ k = -˙ W

Wn

˙ ̇ W = -k ¥ nWn-1 ¥ ˙ W

Substitute for k and tidy up:

†

˙ ̇ W =˙ W

Wn nWn-1 ˙ W

n =W ˙ ̇ W ˙ W 2

Can write this in terms of P and its derivatives as before.Typically n is around (but not exactly equal) to 3.

ASTR 3730: Fall 2003

The pulsar population

period

periodderivative

lines of constantage

pulsar death line

ASTR 3730: Fall 2003

Emission from pulsarsThe optical emission from e.g. the Crab pulsar appears to besynchrotron emission. How does the neutron star produceenergetic particles and magnetic fields?

Rotationaxis Magnetic

axis

Closedmagneticfield lines

Open magneticfield lines

Speed of lightcylinder - whereparticle corotatingwith the neutronstar have v = c

†

v = Wr = c

r =cW

ASTR 3730: Fall 2003

Concept leads to a lighthouse model for pulsars:

Charged particlesare acceleratedoff the surface ofthe neutron starby strong electricfields

Radiation is beamed along directions of open field lines

See pulses of radiation when the beam sweepspast ourdirection